COT3100.01, Fall 2002
Solution Keys to Final Exam (Test #3)
12/03/2002
S. Lang
(12/03/2002)
Part I.
(36 pts., 3 pts. each) True/False, or shortanswer, questions. (No explanation needed and
no partial credit given.)
1.
Let
A
= {1, 2, 3, 4} and let
R
⊆
A
×
A
denote a binary relation depicted by the directed graph given
in the figure.
Answer each of the following True/False questions:
2.
Define two realvalued functions
f
: [0,
∞
)
→
R
and
g
:
R
→
R
by the following formulas, where the
set [0,
∞
) denotes the set of nonnegative real numbers, and
R
denotes the set of all real numbers,
f
(
x
) =
x

1, and
g
(
x
) = 2
x
+ 1.
Answer the following questions:
(a) Is function
f
an injection?
True
, because if
f
(
x
) =
f
(
y
), then
x

1 =
y

1. Adding 1 to both
sides and squaring yields
x
=
y
, which proves the injection property.
(b) Is the inverse of
g
given by the formula
g

1
(
y
) =
1
2

y
?
False
, to find the inverse of function
g
,
let
y
=
g
(
x
) = 2
x
+ 1, then solve for
x
in terms of y.
Thus,
x
=
2
1

y
, which means
g

1
(
y
) =
2
1

y
.
(c) Give the value of
g
ο
f
(4).
Answer
:
g
ο
f
(4) =
g
(
f
(4)) =
g
(
)
1
4

=
g
(1) = 2 + 1 = 3.
(d) Give (an exact description of) the range of
f
.
Answer
: [

1,
∞
) = the set of real numbers that are
≥

1.
3.
Let
a
and
b
denote positive integers.
Answer the following questions:
(a) If an integer
c

a
and
c

b
, then
c
 GCD(
a
,
b
).
True
, because GCD(
a
,
b
) =
at
+
bu
for some
integers
t
and
u
, and c  (
at
+
bu
) if
c

a
and
c

b
are true as given in the assumption.
(b) If both
a
and
b
are prime numbers, then GCD(
a
,
b
) = 1.
False
, for example, GCD(2, 2) = 2.
1
2
4
3
(a)
Is
R
antisymmetric?
True
, because using the
definition, whenever an edge (
a
,
b
)
∈
R
and
a
≠
b
, then
the edge (
b
,
a
)
∉
R
is true.
(b)
Is
R
transitive?
False
, because both (2, 4)
∈
R
and
(4, 3)
∈
R
but (2, 3)
∉
R
.
(c)
Does the pair (1, 2)
∈
R
ο
R
?
True
, because (1, 2) is
the composition of (1, 1) and (1, 2) both of which
∈
R
.
(d)